I've been given the following question:

Let be a real Hilbert space with an orthonormal basis . Let be continuous. Show that there exists a positive self-adjoint operator such that for all .

Show moreover that is compact.

(you may use that as )

I don't quite understand what must be shown here exactly. Must I find such a operator T explicitly and then show it has the desired property. If so, how? I don't quite see where the hint is useful, I guess it's meant for showing compactness of T.

Any hint or push in the right direction is very much appreciated.